Line bundles with smooth metrics on smooth complex analytic spaces allow to perform differential calculus. Namely, the curvature of a smooth metrized line bundle $\overline{L}$ is a differential form $c_{1}(\overline{L})$ of type $(1,1)$ on $\mathrm{X}$. Its definition involves the differential operator

$$\mathop{\mathrm{d}\mathrm{d}^{c}}=\frac{i}{\pi}\partial\overline{\partial}.$$

When an open set $\mathrm{U}\subset\mathrm{X}$ admits local coordinates $(z_{1},\dots,z_{n})$, and $s\in\Gamma(\mathrm{U},L)$ is a local frame, then

$$c_{1}(\overline{L})|_{\mathrm{U}}=\mathop{\mathrm{d}\mathrm{d}^{c}}\log\left\|{s}\right\|^{-1}=\frac{i}{\pi}\sum_{1\leq j,k\leq n}\frac{\partial^{2}}{\partial z_{j}\partial\overline{z}_{k}}\log\left\|{s}\right\|^{-1}\mathrm{d}z_{j}\wedge\mathrm{d}\overline{z}_{k}.$$

Cauchy-Riemann equations ($\partial f/\partial\overline{z}=0$ for any holomorphic function $f$ of the variable $z$) imply that this formula does not depend on the choice of a local frame $s$. Consequently, these differential forms defined locally glue to a well-defined global differential form on $\mathrm{X}$.

Taking the curvature form of a metrized line bundle is a linear operation : $c_{1}(\overline{L}\otimes\overline{M})=c_{1}(\overline{L})+c_{1}(\overline{M})$. It also commutes to pull-back : if $f\colon Y\rightarrow X$ is a morphism, then $f^{*}c_{1}(\overline{L})=c_{1}(f^{*}\overline{L})$.

In the case of the Fubini-Study metric over the projective space ${\mathbf{P}}^{n}({\mathbf{C}})$, the curvature is computed as follows. The open subset $\mathrm{U}_{0}$ where the homogeneous coordinate $x_{0}$ is non-zero has local coordinates $z_{1}=x_{1}/x_{0}$, …, $z_{n}=x_{n}/x_{0}$ ; the homogeneous polynomial $X_{0}$ defines a non-vanishing section $s_{0}$ of $\mathscr{O}(1)$ on $\mathrm{U}_{0}$ and

$$\log\left\|{s_{0}}\right\|^{-1}_{\mathrm{FS}}=\frac{1}{2}\log\left(1+\sum_{j=1}^{n}\left|{z_{j}}\right|^{2}\right).$$

Consequently, over $\mathrm{U}_{0}$,

	$\displaystyle c_{1}(\overline{\mathscr{O}(1)}_{\mathrm{FS}})$	$\displaystyle=\frac{i}{\pi}\partial\overline{\partial}\log\left\|{s_{0}}\right\|^{-1}_{\mathrm{FS}}$
		$\displaystyle=\frac{i}{2\pi}\partial\left(\sum_{k=1}^{n}\frac{z_{k}}{1+\left\|{z}\right\|^{2}}\mathrm{d}\overline{z}_{k}\right)$
		$\displaystyle=\frac{i}{2\pi}\sum_{j=1}^{n}\frac{1}{1+\left\|{z}\right\|^{2}}\mathrm{d}z_{j}\wedge\mathrm{d}\overline{z}_{j}-\frac{i}{2\pi}\sum_{j,k=1}^{n}\frac{z_{k}\overline{z}_{j}}{(1+\left\|{z}\right\|^{2})^{2}}\mathrm{d}z_{j}\wedge\mathrm{d}\overline{z}_{k}.$

In this calculation, we have abbreviated $\left\|{z}\right\|^{2}=\sum_{j=1}^{n}\left|{z_{j}}\right|^{2}$.